Hi
I need help figuring out the set up of this problem.
Jane has 2 tennis serves, a hard serve and a soft serve, her hard serve is good 50% of the time and her soft serve is good 75% of the time. I f her hard serve is in she wims 75% of her points. If her soft serve is in she wins 50% of her points. Since she is allowed to reserve one time if her first serve is out. What should her serving strategy be. That is, should she serve a hard followed by a soft; both hard; soft followed by hard or both soft.

I know that if the first serve is in you play out the point
If the first serve is out you get a second serve
I the second serve is out you lose rhe point
I can't figure out how to set up the problem?

You need to think about it in terms of possible outcomes. For example, suppose she starts with a hard serve. For the first serve, there's a 1/2 chance that she hits the serve. If she hits it, there's a 3/4 chance that she wins (or 1/2 * 3/4 chance that she'll win the serve that way) and 1/4 chance she loses (1/2 * 1/4). If she misses the first serve (already a factor of 1/2 in probability now), then she serves hard again.

Now she has a chance of 1/2 to hit the second serve, and a chance of 3/4 to win if she does. So for the second serve, the probability of winning is 1/2 * 1/2 * 3/4 (prob. That she missed the first serve * prob. That she hits the second one * prob. That she wins it), and a chance 1/2*1/2*1/4 of losing the second serve if she hits it, and a chance 1/2*1/2 of missing both hard serves.

If you add up all those possibilities (3/8 + 1/8 + 3/16 + 1/16 + 1/4), they add up to 1, which tells you that you have all the possibilities covered - one of those outcomes will happen, and the probability that it wills given by the fraction. So to find out the chance of winning, you'd add all the wins up (winning the first serve or winning the second serve).

Then you have 3 other serving situations to consider - hard then soft, soft then hard, and soft/soft. For each set, you'll get a fraction for the total probability that she'll win the point - whichever one's highest is the option to go with.